Israeli President Shimon Peres is trying one last time to nudge forward the peace process.

JERUSALEM—As Pope Francis prepares to enter the fraught world of Middle East peacemaking on Sunday, the Israelis have boycotted talks, the U.S. has called for a pause, and the Palestinians forged a new government backed by militant Islamists.

Amid the breakdown, only one Israeli official is willing to accept the pontiff's invitation to attend a prayer meeting at the Vatican with Palestinian President Mahmoud Abbas: Israeli President...